Involutions of Bicomplex Numbers

An involution of a real commutative algebra $A$ is a real-linear homomorphism $f:A\to A$ such that $f^2=\mathrm{I}\mathrm{d}$. We show that there are six involutions of the algebra of bicomplex numbers, contrary to the actual number of four stated in the literature. We also characterize $n$-involutions satisfying the additional property $f^n=\mathrm{I}\mathrm{d}$ for some integer $n\ge 2$. We show there are eight $n$-involutions and they occur only for $n=2$ and $n=4$. We use our result to give a new characterization of the invertible elements of the algebra of bicomplex numbers.